Step 1: In a large bowl, combine the warm milk, yeast, and 1 teaspoon of the ¼ cup granulated sugar. Stir gently and let sit for 5-10 minutes until foamy. This indicates the yeast is active.
Step 2: To the yeast mixture, add the pumpkin puree, egg, and vanilla extract. Whisk until well combined.
Step 3: In a separate bowl, whisk together the flour, pumpkin pie spice, ½ teaspoon cinnamon, salt, and remaining granulated sugar (from the ¼ cup).
Step 4: Add the dry ingredients to the wet ingredients in three additions, mixing with a wooden spoon or spatula until just combined after each addition. The dough will be very sticky. Pour in the melted butter and mix until just incorporated. Do not overmix.
Step 5: Cover the bowl with plastic wrap and refrigerate for at least 8 hours or up to 24 hours. The dough will slowly rise and become bubbly.
Step 6: The next day, or after chilling, generously flour a clean work surface. Turn the cold dough out onto the floured surface. Lightly flour the top of the dough. Gently roll the dough to about a ½-inch thickness.
Step 7: Using a 3-inch doughnut cutter (or two concentric cookie cutters), cut out doughnuts and doughnut holes. Place the cut doughnuts and holes on parchment-lined baking sheets. Gently re-roll any scraps to cut additional doughnuts.
Step 8: Loosely cover the doughnuts with plastic wrap or a clean kitchen towel and let them rise at room temperature for 30-45 minutes, or until noticeably puffy.
Step 9: While the doughnuts are rising, prepare the cinnamon sugar coating: In a shallow dish, whisk together the ½ cup granulated sugar and 2 teaspoons ground cinnamon. Set aside.
Step 10: Heat 6-8 cups of vegetable oil in a large heavy-bottomed pot or Dutch oven over medium heat until it reaches 350-360°F (175-180°C). Use a thermometer to monitor the temperature.
Step 11: Carefully lower 2-3 doughnuts (do not overcrowd the pot) into the hot oil. Fry for 1-2 minutes per side, or until golden brown and cooked through. Remove with a slotted spoon or spider and place on a wire rack set over paper towels to drain excess oil.
Step 12: While the doughnuts are still warm, but cool enough to handle, gently toss them in the prepared cinnamon sugar mixture until fully coated. Repeat with remaining doughnuts and doughnut holes. Serve immediately and enjoy your delicious No-Knead Pumpkin Doughnuts with Cinnamon Sugar!